On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results (Q2790738)

Let \(L/K\) be a finite Galois extension of number fields with group \(G\) and let \(r\) be an integer. When specialized to the pair \((h^0(\mathrm{Spec}(L))(r),\mathbb Z[G])\), the equivariant Tamagawa number conjecture \(\mathrm{ETNC}(L/K,r)\) predicts the vanishing of a certain element \(T\Omega(\mathbb Q(r)_L,\mathbb Z[G])\) in the relative algebraic \(K\)-group \(K_0(\mathbb Z[G],\mathbb R)\). This element relates the leading terms at \(s=r\) of Artin \(L\)-functions to certain natural arithmetic invariants. In the case that \(T\Omega(\mathbb Q(r)_L,\mathbb Z[G])\) belongs to \(K_0(\mathbb Z[G],\mathbb Q)\) (when \(r=0\), this is equivalent to Stark's conjecture for \(L/K\)), it follows from the canonical isomorphism \(K_0(\mathbb Z[G],\mathbb Q)\cong\bigoplus K_0(\mathbb Z_p[G],\mathbb Q_p)\) that the global ETNC breaks down into local \(\mathrm{ETNC}_p\)'s at all primes \(p\). In this article, for \(r\leq 0\) and a prime \(p\), the authors prove many new cases of the \(p\)-part of the ETNC, as well as unconditional results on the annihilation of classical Galois modules for a large class of interesting extensions.NEWLINENEWLINEMore specifically, denote by \(DT(\mathbb Z_p[G])\) the torsion subgroup of \(K_0(\mathbb Z_p[G],\mathbb Q_p)\). For any normal subgroup \(N\) of \(G\), consider the natural quotient map \(DT(\mathbb Z_p[G])\to DT(\mathbb Z_p[G/N])\). By studying the structure of the \(p\)-adic group ring \(\mathbb Z_p[G]\), the authors can give criteria for this quotient map to be an isomorphism. If moreover \(T\Omega(\mathbb Q(r)_L,\mathbb Z[G])\) is torsion, the functorial properties of the conjecture show that \(\mathrm{ETNC}_p(L/K,r)\) is equivalent to \(\mathrm{ETNC}_p(L^N/K,r)\). This allows to produce new cases of \(\mathrm{ETNC}_p(L/K,r)\) by reducing to known cases of \(\mathrm{ETNC}_p(L^N/K,r)\). For example, let \(q\) be a power of a prime \(l\) and take \(G\cong\mathrm{Aff}(q)\), the group of affine transformations on \(\mathbb F_q\). Since the complex irreducible characters of \(\mathrm{Aff}(q)\) are either linear or rational-valued, it is known that any Galois extension \(L/\mathbb Q\) with group \(\mathrm{Aff}(q)\) verifies the strong Stark conjecture, which is equivalent to the property that \(T\Omega(\mathbb Q(r)_L,\mathbb Z[G])\) is torsion. Take \(N\cong \mathbb F_ q\) to be the commutator subgroup of \(\mathrm{Aff}(q)\). For any \(p\neq l\), the authors show that \(\mathbb Z_p[G]\) is isomorphic to the direct sum of \(\mathbb Z_p[G/N]\) and some maximal \(\mathbb Z_p\)-order, hence \(\mathrm{ETNC}_p(L/\mathbb Q,0)\) holds by the theorem of \textit{D. Burns} and \textit{C. Greither} [Invent. Math. 153, No. 2, 303--359 (2003; Zbl 1142.11076)] (and \textit{M. Flach} [J. Reine Angew. Math. 661, 1--36 (2011; Zbl 1242.11083)] for \(p=2\)). For \(r<0\) and \(G\cong\mathrm{Aff}(q)\), the authors can prove the validity of \(\mathrm{ETNC}(L/\mathbb Q,r)\) away from the prime 2 when \(L\) is totally real (unconditionally, i.e., without assuming the vanishing of the \(\mu\)-invariant).NEWLINENEWLINESkipping other results on infinite families of non-abelian extensions \(L/\mathbb Q\) satisfying \(\mathrm{ETNC}(L/\mathbb Q,0)\), and on the vanishing of certain elements in relative algebraic \(K\)-groups (epsilon constant conjecture, leading term conjecture at \(s=1\dots\)), let us come to the Galois annihilation conjectures of \textit{D. Burns} [Invent. Math. 186, No. 2, 291--371 (2011; Zbl 1239.11128)] (resp. [the second author, Math. Proc. Camb. Philos. Soc. 151, No. 1, 1--22 (2011; Zbl 1254.11096)]) in the setting of a Galois extension \(L/K\), which are generalizations of the well-known conjectures of Brumer (resp. Coates-Sinnott) on the annihilation of the class group (resp. the higher \(K\)-groups) and which are both implied by \(\mathrm{ETNC}(L/K,r)\). Here the authors prove Burns' conjecture for a wide class of interesting extensions. For example, in the previous case of a Galois extension \(L/\mathbb Q\) with group \(\mathrm{Aff}(q)\), since \(\mathrm{ETNC}_p(L/\mathbb Q,0)\) holds, so does the \(p\)-part of Burns' conjecture for \(p\neq l\). However, by considering certain``denominator ideals'' which play a role in many annihilation conjectures, the \(l\)-part of Burns' conjecture can be deduced (up to the factor 2 if \(l=2\)) from the strong Stark conjecture, even though \(\mathrm{ETNC}_l(L/\mathbb Q,0)\) is not known in this case. The method works equally well in other situations, allowing to prove certain cases of Nickel's conjecture.